Set theory and its applications. L. Babinkostova, A. E. Caicedo, S. Geschke, M. Scheepers, eds. Contemporary Mathematics, vol. 533, Amer. Math. Soc., Providence, RI, 2011. ISBN-10: 0-8218-4812-7 ISBN-13: 978-0-8218-4812-8

Here is a link to the AMS page for it, a link to its table of contents, and the preface:

The Boise Extravaganza in Set Theory (BEST) started in 1992 as a small, locally funded conference dedicated to Set Theory and its Applications. A number of years after its inception BEST started being funded by the National Science Foundation. Without this funding it would not have been possible to maintain the conference. The conference remained relatively small with many opportunities for its participants to meet informally. We like to think that during these years BEST has made it possible for the numerous set theorists who have participated in it to absorb, besides the new developments featured in the conference talks, also part of the folklore and traditions of the ﬁeld of set theory and its relatives. An explicit effort was made to bring together role models from various career stages in set theory as well as the new generation to support some notion of continuity in the ﬁeld.
This volume has a similar purpose. In it the reader will ﬁnd valuable papers ranging from surveys that put in print here set theoretic knowledge that has been around for several decades as unpublished lore, to hybrid survey-research papers, to pure research papers. Readers can be assured of the authority of each paper since each has been carefully refereed. The reader will also ﬁnd that the subjects treated in these papers range over several of the historically strongly represented areas of set theory and its relatives. Rather than expounding the virtues of each paper individually here, we invite the reader to learn from the authors.
Bringing to publication such a collection of papers is not possible without the generous dedication of authors and referees and the services of a publisher. We would like to thank all authors and referees for their selﬂess contributions to this volume. And we particularly would like to thank the publisher, Contemporary Mathematics, and Christine Thivierge, for the guidance they provided during this process.

43.614000-116.202000

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Nice sharing.
Many k-12 and undergraduate students think that Set theory is a basic thing made for children and has no application in advanced mathematics.
But in fact the set theory is the basis of the whole of mathematics.

(As I pointed out in a comment) yes, partial Woodinness is common in arguments in inner model theory. Accordingly, you obtain determinacy results addressing specific pointclasses (typically, well beyond projective). To illustrate this, let me "randomly" highlight two examples: See here for $\Sigma^1_2$-Woodin cardinals and, more generally, the noti […]

I am not sure which statement you heard as the "Ultimate $L$ axiom," but I will assume it is the following version: There is a proper class of Woodin cardinals, and for all sentences $\varphi$ that hold in $V$, there is a universally Baire set $A\subseteq{\mathbb R}$ such that, letting $\theta=\Theta^{L(A,{\mathbb R})}$, we have that $HOD^{L(A,{\ma […]

A Wadge initial segment (of $\mathcal P(\mathbb R)$) is a subset $\Gamma$ of $\mathcal P(\mathbb R)$ such that whenever $A\in\Gamma$ and $B\le_W A$, where $\le_W$ denotes Wadge reducibility, then $B\in\Gamma$. Note that if $\Gamma\subseteq\mathcal P(\mathbb R)$ and $L(\Gamma,\mathbb R)\models \Gamma=\mathcal P(\mathbb R)$, then $\Gamma$ is a Wadge initial se […]

Craig: For a while, there was some research on improving bounds on the number of variables or degree of unsolvable Diophantine equations. Unfortunately, I never got around to cataloging the known results in any systematic way, so all I can offer is some pointers to relevant references, but I am not sure of what the current records are. Perhaps the first pape […]

Yes. Consider, for instance, Conway's base 13 function $c$, or any function that is everywhere discontinuous and has range $\mathbb R$ in every interval. Pick continuous bijections $f_n:\mathbb R\to(-1/n,1/n)$ for $n\in\mathbb N^+$. Pick a strictly decreasing sequence $(x_n)_{n\ge1}$ converging to $0$. Define $f$ by setting $f(x)=0$ if $x=0$ or $\pm x_n […]

One way we formalize this "limitation" idea is via interpretative power. John Steel describes this approach carefully in several places, so you may want to read what he says, in particular at Solomon Feferman, Harvey M. Friedman, Penelope Maddy, and John R. Steel. Does mathematics need new axioms?, The Bulletin of Symbolic Logic, 6 (4), (2000), 401 […]

"There are" examples of discontinuous homomorphisms between Banach algebras. However, the quotes are there because the question is independent of the usual axioms of set theory. I quote from the introduction to W. Hugh Woodin, "A discontinuous homomorphism from $C(X)$ without CH", J. London Math. Soc. (2) 48 (1993), no. 2, 299-315, MR1231 […]

This is Hausdorff's formula. Recall that $\tau^\lambda$ is the cardinality of the set ${}^\lambda\tau$ of functions $f\!:\lambda\to\tau$, and that $\kappa^+$ is regular for all $\kappa$. Now, there are two possibilities: If $\alpha\ge\tau$, then $2^\alpha\le\tau^\alpha\le(2^\alpha)^\alpha=2^\alpha$, so $\tau^\alpha=2^\alpha$. In particular, if $\alpha\g […]

Fix a model $M$ of a theory for which it makes sense to talk about $\omega$ ($M$ does not need to be a model of set theory, it could even be simply an ordered set with a minimum in which every element has an immediate successor and every element other than the minimum has an immediate predecessor; in this case we could identify $\omega^M$ with $M$ itself). W […]

The study of finite choice axioms is quite interesting. Besides the reference given in Asaf's answer, there are a few papers covering this topic in detail. If you can track it down, I suggest you read MR0360275 (50 #12725) Reviewed. Conway, J. H. Effective implications between the "finite'' choice axioms. In Cambridge Summer School in Mat […]

Nice sharing.

Many k-12 and undergraduate students think that Set theory is a basic thing made for children and has no application in advanced mathematics.

But in fact the set theory is the basis of the whole of mathematics.

Are the BEST conferences still going on?

Hi Aaron. I expect so. We skipped a year.